arXiv:math/0212214v5 [math.AG] 26 Apr 2006

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Stability conditions and the braid group

R. P. Thomas

Abstract. We find stability conditions ([D2], [Br1]) on some derived categoriesof differential graded modules over a graded algebra studied in [RZ], [KS]. Thiscategory arises in both derived Fukaya categories and derived categories of coher-ent sheaves. This gives the first examples of stability conditions on the A-modelside of mirror symmetry, where the triangulated category is not naturally thederived category of an abelian category. The existence of stability conditions,however, gives many such abelian categories, as predicted by mirror symmetry.

In our examples in 2 dimensions we completely describe a connected com-ponent of the space of stability conditions. It is the universal cover of the con-figuration space C

0k+1 of k + 1 points in C with centre of mass zero, with deck

transformations the braid group action of [KS], [ST]. This gives a geometricorigin for these braid group actions and their faithfulness, and axiomatises theproposal for stability of Lagrangians in [Th] and the example proved by meancurvature flow in [TY].

1. Introduction

This paper presents a result in pure algebra, but one which is motivated en-tirely by geometry and physics, especially mirror symmetry. It gives examples ofDouglas and Bridgeland’s notion of stability conditions for triangulated categories[D1], [D2], [AD], [Br1], and draws together and axiomatises many of the knowntests of Kontsevich’s homological mirror conjecture [K] (for instance on stabilityof Lagrangians in Fukaya categories [Th], Shapere-Vafa’s examples of special La-grangians [SV], [TY], and braid groups of monodromies [KS], [ST]). We explainsome of the geometry and physics background, distil this into a purely algebraicsetup, and then apply the axioms of stability conditions [Br1] to this problem. Theresult, which can be read independently of the previous sections, is a description of(a connected component of) the space of stability conditions on a certain naturaltriangulated category arising in many areas of geometry and algebra [RZ], [KS],giving both an axiomatic justification for the conjectures and results of [Th], [TY],and a geometric “explanation” or origin for the faithful braid group actions of [KS],[ST], at least in dimension 2. There are other braid group actions in [Sz]; it wouldbe nice to see them arise from stability conditions also.

1

http://arxiv.org/abs/math/0212214v5

2 R. P. THOMAS

Acknowledgements. This project started out as joint work with Tom Bridgeland,who pulled out due to laziness and my misuse of the term “Jordan-Holder filtration”.I owe him a great deal for discussions, an advanced copy of his axioms [Br1], andthe many errors that he spotted. He has now produced a much more professionalpaper [Br2] extending the results of this paper to all Dynkin diagrams. I would alsolike to thank Ludmil Katzarkov and Paul Seidel for extremely useful conversations.The author is supported by a Royal Society university research fellowship.

2. Geometry and physics background

Consider the following much-studied ([KS], [SV], [TY]) affine algebraic varietyX = XN (p):

(2.1)

N∑

i=1

x2i = p(t)

⊆ C

N ×C,

where p is some degree k+1 polynomial in t ∈ C with only simple zeros. There is anatural Kahler form ω restricted from CN+1, and a nowhere-zero holomorphic vol-ume form Ωp (“almost Calabi-Yau” structure) given by taking the Poincare residue

([GH] p 147) of the meromorphic form dx1 . . . dxN dt/(∑

x2i − p(t))on CN+1; this

can be written as

(2.2) Ω = Ωp = (−1)N+i+1 dx1 . . . dxi . . . dxN dt|XN

2xi=dx1 . . . dxN |XN

p(t)

for any i (so where xi = 0 ∀i we can use the second expression). Here dxi meansthat we omit the dxi term from the wedge product.

Each smooth fibre over t ∈ C is an affine quadric with a natural LagrangianSN−1 “real” slice, namely the intersection of the fibre with the slice

xi ∈√p(t)R ∀i.

It is invariant under the obvious O(N) action on XN , and is the vanishing cycleof every singular fibre (i.e. the fibres over the roots of p). Therefore any pathγ : [0, 1] → C from one zero of p to another (avoiding zeros of p in its interior) liftsto give a canonical O(N)-invariant Lagrangian N -sphere, SN−1-fibred over γ exceptat the endpoints where it closes up.

Picking an ordering of the k + 1 zeros of p, we can pick k consecutive pathsin C joining one zero to the next, forming a so-called “Ak-chain” of paths. Thecorresponding Lagrangian N -spheres Li

ki=1 in XN (the vanishing cycles of the

Ak-singularity; the singularity obtained by putting p(t) = tk+1 in (2.1)) form abasis for its homology and define an Ak-chain of Lagrangians [Se1]. That is, theirgeometric intersections satisfy

|Li.Lj | = 1, |i− j| = 1, |Li.Lj | = 0, |i− j| > 1.

STABILITY CONDITIONS AND THE BRAID GROUP 3

Choosing their gradings [Se2] appropriately, we can arrange that their Floer coho-mologies satisfy

HF 0(Li, Li) = C = HFN (Li, Li),

HF 1(Li, Li+1) = C = HFN−1(Li+1, Li),(2.3)

for all i, with all other groups zero.As p varies, all of these varieties Xp are isomorphic as symplectic manifolds,

even though as complex manifolds they are varying. Thus around loops in thespace of polynomials with simple zeros p we get monodromy in the symplecticautomorphism group Aut(X,ω). Scaling p makes no difference to the symplecticgeometry, nor, in dimension N = 2 to the graded symplectic geometry, as Ωp (2.2)is left unaltered. Similarly everything is invariant under translations in the t-plane;dividing by these reparametrisations leaves us with the (simple) roots of p normalised(say) to have centre of mass zero. That is we get the configuration space C0

k+1 ofk + 1 points of mean zero in C, with fundamental group the braid group Bk. Thisgives a representation of Bk → Aut(X,ω), generated by “generalised Dehn twists”about the vanishing cycles; this amounts to “twisting” about the Lagrangian fibredover a path between two roots of p by rotating the two points in C about eachother; i.e. it arises from the usual description of Bk lifted from C (with k+1 markedpoints) to X. In fact these Dehn twists give such a braid group action on anysymplectic manifold with an Ak-chain of Lagrangian spheres [Se1], and so thisa local model for symplectic automorphisms of compact symplectic manifolds, inparticular Calabi-Yau manifolds. This induces a braid group action on the derivedFukaya category Db(Fuk(X)) of X (though one must check first, as in [Se2], thatthe symplectomorphisms lift naturally to the graded symplectomorphism group).

Under mirror symmetry (and there is a proposal for the mirror of this Ak-smoothing in [ST] Section 3f) we cannot expect a braid group action of holo-morphic automorphisms of any mirror complex manifold Y ; there are in generalvery few holomorphic automorphisms of Calabi-Yau manifolds, and it is one ofKontsevich’s great insights that the mirror is really the bounded derived categoryDb(Y ) ∼= Db(Fuk(X)) of coherent sheaves on the Calabi-Yau rather than the Calabi-Yau itself; in this way the automorphisms can be matched, so things work in families.

So if the only vestiges of our geometric picture above that remain under themirror map are categorical, we would like to see the varying complex structure(given by the polynomial p) that induces the braid monodromy at the level of the(purely symplectically defined) derived Fukaya category. How one “ought” to dothis has long be known to physicists, and was described to me many years agoby Eric Zaslow. Namely one should follow the D-branes in Db(Fuk(Y )); thesedepend on the complex structure, and are thought to be the special Lagrangians(Lagrangians L for which the N -form (2.2) Ω|L has constant phase on L). Ongoing round a loop in complex structure moduli space the set of special Lagrangiansundergoes monodromy which one might hope extends uniquely to the full derivedcategory. Similarly in the mirror picture the D-branes in Db(X) depend on theKahler structure, and are to a first approximation the stable bundles that satisfy anappropriately perturbed Hermitian-Yang-Mills equation. In [Th] this picture was

4 R. P. THOMAS

mirrored to give an appropriate notion of stability for Lagrangians which shouldconjecturally be equivalent to the Lagrangian being a hamiltonian deformation of a(unique) special Lagrangian. In the case of the Lagrangians of our above example,this can all be illustrated very simply [TY], at least in N = 2 dimensions (higherdimensions are more complicated, but also illustrated in [TY]). O(N)-invariantgraded [Se2] Lagrangian spheres correspond to paths in C between distinct zeros ofp(t), missing other zeros, with a continuous choice of lift to R of the phase ∈ R/2πZof the tangent to the path at each point. Special Lagrangians correspond to (spheresfibred over) straight lines, and O(N)-invariant hamiltonian isotopies to isotopies ofthe path in C, with endpoints fixed, not crossing any other zeros of p. Extensionscorrespond to graded Lagrangian connect sums #, and stability of L to there beingno such connect sum L1#L2 hamiltonian isotopic to L, with the phase of L1 greaterthan that of L2; see Figure 1, in which L1 has phase ±ǫ and L2 has phase zero.

0

ǫ

L2 L2

L1

L1

−ǫ

0C

A

D

B

Figure 1. The connect sums L1#L2 (A – stable, B – unstable) andL2#L1 (C – unstable, D – stable) in 2-dimensions, in 2 differentcomplex structures.

Simple examples of loops of complex structures p do indeed exhibit the braidmonodromy action on the set of such special Lagrangians [TY]. For a long timeit was not clear how to extend such naive notions of stability (in terms of injectivemorphisms or subobjects) to full triangulated categories (where there is no notionof subobject). But the beautiful work of Douglas [D2], now axiomatised in [Br1],purports to give the answer; the purpose of this paper is to show how it works in ourAk example, and how it agrees with the naive definitions mentioned above. This alsogives a new example where the axioms of [Br1] hold; in general examples are hard tofind. Known examples work on the B-model side where there is an abelian category(the coherent sheaves), and where the usual notions of (semi)stability of sheaves arewell understood and give examples satisfying the axioms (with the semistable objectsof the axioms below being the semistable sheaves and their translations) which canbe deformed to give more. In the example of this paper there is no natural abeliancategory, as we start on the A-model side with the Fukaya category. Using the stableobjects of [TY] we prove the existence of stability conditions, which in turn showsthat the derived Fukaya category is the derived category of many abelian categories.

Our results also apply to the B-model derived category of coherent sheaves, andthus gives a geometric, monodromy background for the mirror braid group action


on the derived category described in [ST]. The varying complex structure p is theninterpreted as a varying complexified Kahler form on the mirror manifold containingan Ak-chain of spherical objects. In fact we do not use p, but the closely related setof the values it gives to the Ak-chain of Lagrangians via (2.2):

(2.4)

i∑

j=1

Z(Lj) =

∫

L1∪...∪Li

Ωp

k

i=1

.

In two dimensions Z(Li) is a constant times by the vector in C between the two zerosof p that form the endpoints of the path over which Li is S

1-fibred (see (A.1) of [Ho]).Thus, up to an additive constant in C which does not affect the homotopy type, thenumbers (2.4) and the origin together constitute the distinct roots of p in C, and themap p 7→

0,∑

j≤i Z(Lj)is a C∗-bundle inducing an isomorphism π1/π1(C

∗) →

π1(Ck+1) ∼= Bk. (In higher dimensions the relationship is more complicated; forsmall paths, Z(Li) is roughly the (n/2)th power of the vector in C representing Li,which is why it is simpler to stick to two dimensions for our analysis of monodromy.)Instead of the space of p s, then, we will find the space of stability conditions to bethe universal cover of the configuration space C0

k+1 of (k+1) distinct points of meanzero in C, with fundamental group Bk.

Most of the work can be done by just dealing with curves in the plane (withendpoints in the k + 1 marked points) as in [TY], and the stability conditions thatemerge axiomatise that of [Th], [TY]; i.e. relate to whether or not the path can bepulled “tight” (straight, in the dimension N = 2 case, or to a special Lagrangian of[SV] in general; this was done in [TY] by mean curvature flow) without touching oneof the other marked points. Unfortunately one has to deal with slightly more generalobjects in Db(Fuk(X)) than can be represented solely by curves in the plane. Thisis because the curves do not form an abelian, or triangulated, category – in generalone cannot form the kernel or cokernel or cone on a morphism (element of Floerhomology) between two Lagrangians. They do form an A∞ category in a complicatedway (which involves difficult counting of holomorphic discs with boundaries in theLagrangians), though an intrinsic formality result [ST] means we will not haveto worry about the precise A∞ structure. We then have to derive this category,a formal procedure producing cones on abstract morphisms that introduces extraobjects not all representable by curves. As proposed in [Th], Ext1s are geometricallyrepresented by (graded) connect sums of Lagrangians; what stops us from using thisto form geometric representatives of all cones is the fact that some of the Ext1s leadto immersed Lagrangians, whose Floer homology is not well understood, and somelead to different representatives of the same class in the Fukaya category if one eithertakes the relative connect sum [Th] or perturbs and takes a transverse connect sum.It would be nice to find a purely one-dimensional geometric description of all objectsof this category.

Nonetheless our constructions are motivated by pictures for those objects ofDb(Fuk) that are representable by curves, and so such figures often accompanyalgebra below without any explanation; these should be helpful to anyone who hasread the above background. Unfortunately such physical arguments do not suffice

6 R. P. THOMAS

for the whole category for the A∞ reasons mentioned above, but nonetheless givea good intuitive picture for the axioms [Br1] that will be familiar to physicists,reminiscent of “marginal stability”.

To show that our results are relevant to “real life” we now define our category andshow it is indeed faithfully included in the derived Fukaya category of any symplecticmanifold with an Ak-chain of Lagrangian spheres, and in the derived category of anysmooth projective variety with an Ak-chain of spherical objects [ST].

3. The categories

We start by defining a simple auxiliary category to define our graded algebra(closely related to the one studied in [RZ], [KS]) whose derived category of dgmswill be the triangulated category of study. Compare (2.3).

Definition 3.1. Let C = CNk be the C-linear graded category of an Ak-chain

Ejkj=1 in dimension N ; the category with k spherical objects Ej ,

• Homi(Ej , Ej) =

C i = 0, N0 otherwise,

and morphisms

• Homi(Ej , Ej+1) =

C i = 10 otherwise

• Homi(Ej+1, Ej) =

C i = N − 10 otherwise

• Homi(Ej , Ek) = 0, |j − k| > 1.

Denoting by (j, j±1) the generator of Homi(Ej , Ej±1), we impose the relation that

(j, j±1) (j±1, j) is the generator fj of HomN (Ej , Ej) for all j; in particular then(j, j+1)(j+1, j) = (j, j−1)(j−1, j) = fj, and the category has a duality: the pairing

Homi(F,E) ⊗HomN−i(E,F ) → HomN (E,E) ∼= C is perfect for all E,F ∈ C.

In the usual way C defines a graded unital algebra End C:

Ak = ANk = End

(⊕

j

Ej

)=

⊕

ijk

Homi(Ej , Ek).

This is a quotient of the path algebra of the quiver in Figure 2. Generators, over

k

1 1

N - 1 N - 11 2 3

Figure 2.

C, of the algebra are given by oriented paths, graded as in the Figure, with multi-plication given by composition of paths (this is the dual picture to the Lagrangian


one described in the introduction, with vertices in Figure 2 corresponding to La-grangians, i.e. paths between zeros of p, and arrows corresponding to intersectionsof Lagrangians, i.e. zeros of p). The quotient is by the two sided ideal generated by(i, i − 1, i) − (i, i + 1, i), (i − 1, i, i + 1) and (i + 1, i, i − 1) for all i = 2, . . . , k − 1,in the obvious notation; see ([ST] Section 4c) for more details. Denote by ei theidentity in Hom(Ei, Ei) (i.e. the constant path at the ith node), so that 1 =

∑i ei

and left (right) multiplication by ei is projection onto those paths that begin (end)at the ith node. Similarly we denote fi = (i, i − 1, i) = (i, i + 1, i) (or whichever ofthe two is defined, if i = 1 or k).

Now form the bounded derived category D(Ak) of differential graded (right)modules (dgms) over Ak; again see ([ST] Section 4a). This is not the derivedcategory of the abelian category of dgms over Ak, but the localisation of this categoryby quasi-isomorphisms (dgm maps that induce isomorphisms on the cohomologygraded modules). It is, however, triangulated. Denote by Pi the projective (right)module eiAk; these form an Ak-chain in D(Ak) in the sense of Definition 3.1 (withHomi the ith cohomology of Hom) and so form a subcategory whose cohomologycategory is isomorphic to the original Ck.

Definition 3.2. Dk = DNk is the sub-triangulated category of D(Ak) generated

by the Pi; the smallest triangulated subcategory containing the Pi.

Note that the duality (3.1) induces a “trace map” HomN (E,E) → C for allE ∈ Dk such that there exists a perfect pairing Homi(F,E) ⊗ HomN−i(E,F ) →HomN (E,E) → C, i.e. a duality

(3.3) HomN−i(E,F )∗ ∼= Homi(F,E),

functorial in E and F , for all E,F ∈ Dk.This is the category whose stability conditions we will study in this paper. Before

we do, we point out that due to the intrinsic formality result of [ST], it is contained inthe derived Fukaya category of a symplectic manifold containing a (suitably graded)Ak-chain of Lagrangian spheres, and, mirror to this, in any derived category ofcoherent sheaves containing a (suitably shifted) Ak-chain of spherical objects Ei[ST]. Many thanks to Paul Seidel for this argument.

In the first case, the full subcategory A defined by the Ak-chain defines a naturaldga whose cohomology algebra is our AN

k . By intrinsic formality [ST], then, A

is actually quasi-isomorphic to ANk . Since the derived Fukaya category contains

cones, Tw(A), the triangulated category of twisted complexes [BK] on A, is a fullsubcategory of the derived Fukaya category. But it is quasi-equivalent to Tw(AN

k ),which is equivalent to the derived category of dgms generated by the projectivemodules Pi = eiA above (since the functor Hom( · ,⊕iEi) from CN

k to the category

of dgms over ANk = Hom(⊕iEi,⊕iEi) is full and faithful by the Yoneda lemma, and

takes the objects Ei of CNk to the projectives Pi).

Similarly, given an Ak-chain in a derived category of coherent sheaves over asmooth projective variety, we can work with the equivalent homotopy category ofcomplexes of quasicoherent injective sheaves with bounded coherent cohomology.This is a dg-category containing cones, so is isomorphic to its own Tw. The Ak-chain

8 R. P. THOMAS

defines a full sub-dg-category defining a dga quasi-isomorphic to its cohomologygraded algebra AN

k by intrinsic formality. The rest of the argument is then thesame.

Proposition 3.4. For N ≥ 2, DNk is fully faithfully embedded in the derived

Fukaya category of any 2N -dimensional symplectic manifold containing an Ak-chainof spheres; similarly for the derived category of coherent sheaves on any smoothquasiprojective N -dimensional variety with an Ak-chain of spherical objects.

For instance in dimension two we can consider the coherent sheaves over C 2

(finitely generated modules over C [x, y]) supported at the origin, and the standard

SU(2) action of Z/k on C 2. Then the derived category DZ/k0 (C 2) of equivariant

sheaves supported at the origin is equivalent to the derived category of coherent

sheaves on the minimal resolution C 2/(Z/k) supported on the exceptional set E[KV], [BKR]. The exceptional locus E is an Ak-chain of −2-spheres whose structuresheaves form an Ak-chain in the derived category [ST], and correspond under theequivalence to the non-trivial irreducible representations of Z/k. So D2

k is embedded

in DZ/k0 (C 2) ∼= Db

E

(C 2

Z/k

).

4. Stability

In [Br1], a notion of stability for derived categories is given, axiomatising theproposal of Douglas [D2]. Let K(T ) denote the Grothendieck group of T .

Definition 4.1. A stability condition (Z,SS) on a triangulated category Tconsists of a linear map Z : K(T ) → C and full subcategories SS(φ) ⊂ T for eachφ ∈ R satisfying the following five axioms:

(a) for all φ ∈ R, SS(φ+ 1) = SS(φ)[1],(b) if E ∈ SS(φ) then Z(E) = m(E) exp(iπφ) with m(E) > 0,(c) for 0 6= E ∈ T there is a finite sequence of real numbers

φ1 > φ2 > · · · > φn

and a collection of triangles

0 E0// E1

//

E2//

. . . // En−1// En

E

S1

^^<<

<<

S2

^^<<

<<

Sn

``AA

AA

with Si ∈ SS(φi) for all i,(d) if φ1 > φ2 and Si ∈ SS(φi) then HomT (S1, S2) = 0,(e) the subset

Z( ⋃

φ∈R

SS(φ))⊂ C

has no limit points in C.


The map Z is called the central charge of the stability condition. The objects ofthe subcategory SS(φ) are said to be semistable of phase φ; the simple semistablesare stable, and we denote these by S(φ). We call the choice of a lift of the phaseeiπφ of such an object E to a real number φ a grading of E. It is an easy exerciseto check that the decomposition of a nonzero object E given in (c) is unique; theobjects Si are called the semistable factors of E. We sometimes call such a collectionof triangles a filtration. The mass of E is the positive real number

(4.2) m(E) =∑

i

|Z(Si)|.

By the triangle inequality one has m(E) ≥ |Z(E)| with equality if E is semistable.The axioms are modelled on semistability for sheaves on complex curves: on

filtering objects by their cohomology sheaves, and these in turn by their Harder-Narasimhan filtrations, we get an example satisfying axiom (c) above.

To make this definition more manageable we give some conditions that will implythe difficult axiom (c).

Theorem 4.3. Given (T , Z,S(φ)), and defining SS(φ) to consist of all possibleextensions of elements of S(φ), suppose that these satisfy axioms (a,b,d,e) above.Suppose that

⋃φ S(φ) = Si[m] : i = 1, · · · , k, m ∈ Z, for some finite set of

Si which generate T (i.e. every object in T is a finite extension of shifts of Sis).Suppose also that for any non-trivial element of Ext1(F,E) defining a triangle

(4.4) E → C → F, with E ∈ S(φ), F ∈ S(ψ), φ < ψ,

C is either in S(θ) with θ ∈ (φ,ψ), or C = A⊕ B, with A ∈ S(α), B ∈ S(β), φ <α ≤ β < ψ. Then (T , Z,SS(φ)) satisfy axiom (c), i.e. they define a stability condi-tion.

Remark. This theorem is surely true more generally: that we get a stability condi-tion if all extensions (4.4) have a Harder-Narasimhan filtration by shifts of Sis (ratherthan just splitting into two stable objects), but we will only require the above result.

Proof. We have to find a Harder-Narasimhan filtration (c) for any object E ∈ T .Since the Si generate T , we can find a collection of triangles

0 E0// E1

//

E2//

. . . // En−1// En

E

Q1

^^<<

<<

Q2

^^<<

<<

Qn

``AA

AA

(4.5)

with each Qj ∈⋃

i,mSi[m] stable of phase φj .Suppose that for some i, φi < φi+1. Replace Ei−1 → Ei → Ei+1 in the above

filtration by the composition Ei−1 → Ei+1; this has cone Q fitting into a triangle

(4.6) Qi → Q→ Qi+1.

We can now use the assumption on such extensions, as φi < φi+1 and the Qi arestable.

10 R. P. THOMAS

Either (i) Q is stable, and we replace (4.5) by our new filtration with one lesstriangle (with Qi, Qi+1 replaced by Q of phase φ ∈ (φi, φi+1)). We then start theprocess again, looking for j such that φj < φj+1.

Or (ii) Q = 0, so Qi+1 = Qi[1]; in this case we also remove Ei+1 from thefiltration to give a new filtration with Ei−1 → Ei+2 in the middle (with cone Qi+2

forming the new triangle).Or (iii) Q = Q′

i ⊕ Q′i+1 is the direct sum of two stables of phases φ′i ≥ φ′i+1

(without loss of generality) in [φi, φi+1] (with the closed interval being necessary incase the extension (4.6) is the trivial one). In this case we define E′

i by the triangle

E′i → Ei+1 → Q′

i+1,

where the second arrow is the composition of Ei+1 → Q and Q = Q′i⊕Q

′i+1 → Q′

i+1.Then a small check with the octahedral Lemma gives us a new filtration (4.5) withEi−1 → Ei → Ei+1 replaced by

Ei−1// E′

i//

Ei+1

~~||||

||||

Q′i

__>>

>>

Q′i+1

__>>

>>

Again we now start again with this filtration.We claim this procedure terminates; that is after a finite number of steps we

have that φi ≥ φi+1 for all i. To demonstrate this, we assign to each such filtra-tion a number which is both bounded below and decreases (by some bounded belowamount) at each stage. Firstly, we may assume without loss of generality (by re-placing E by E[k] for some k if necessary) that each phase φi of the Qi in (4.5) ispositive.

Then to (4.5) we can associate the real number∑n

k=1 f(k)φk for some strictlypositive, strictly increasing, concave function f(x). Then in case (i) above thisclearly decreases, as φ < φi+1, so that f(i)φ < f(i+1)φi+1 < f(i)φi + f(i+1)φi+1,and the sum over all higher k ≥ i + 1 is also smaller. Case (ii) is even clearer(remembering that all φi > 0). Finally for case (iii) we pick a sufficiently concavefunction f such that

f(x)φ+ f(x+ 1)ψ > f(x)β + f(x+ 1)α,

for all x ≥ 0 and φ < α ≤ β < ψ coming from phases of extensions of stable objectsas in the assumptions of the Theorem; this is possible since the number of suchextensions is finite (up to shifts, which leave the above inequality unaffected).

Then in case (iii) f(i)φi + f(i + 1)φi+1 > f(i)φ′i + f(i + 1)φ′i+1 (i.e. the aboveinequality with φ′i = β, φ′i+1 = α) ensures that the functional again decreases.

This procedure now terminates as the amount the functional decreases by isbounded below by the discreteness of the phases of the stable objects. This givesus a “Jordan-Holder filtration” of E into stable objects of nonincreasing phases (weuse the term as in holomorphic bundle theory or [TY] for Lagrangians, rather thanthe category theoretic terminology). To get the Harder-Narasimhan filtration (c)we bundle together any Qis of the same phase. That is, if φi = φi+1, replace the


Ei−1 → Ei → Ei+1 part of the filtration by just Ei−1 → Ei+1, with one less trianglewith cone Q fitting into a triangle

Qi → Q→ Qi+1.

By assumption this means Q is semistable of phase φi = φi+1; now continue theprocess until the phases are strictly decreasing.

We mention in passing that given a stability condition we get a family of boundednondegenerate t-structures Ft = Ft+1[−1] on T given by the full subcategory ofobjects whose semistable factors (c) all of phase > t [Br1]. This has as its heart

(4.7) Ft ∩ F⊥t+1,

the full subcategory of objects whose semistable factors (c) all have phase in (t, t+1],so in particular we can assign a phase φ(A) ∈ (t, t + 1] to each object A 6= 0 of the

heart such that Z(A) = m(A)eiπφ(A) with m(A) > 0. It turns out then [Br1] that Ais semistable if and only if for every subobject B of A in the heart (which is, recall,an abelian category), φ(B) ≤ φ(A). We may also take B to be stable in this test.We also get (a more standard) Harder-Narasimhan filtration (c) of objects in theheart by semistable objects of the heart.

5. Our example

We first need some technical results about our projective modules Pi in DNk .

Given modules A, B and an element e of Ext1(B,A) = Hom0(B[−1], A), we willoften denote by A#B the corresponding extension (i.e. the cone on B[−1] → A)fitting into the triangle

B[−1] → A→ A#B → B.

This defines a canonical e ∈ Hom0(A#B,B).

Lemma 5.1. Given e ∈ Ext1(C,A) and f ∈ Ext1(C,B) defining A#C, B#C

and e ∈ Hom0(A#C,C), f ∈ Hom0(B#C,C), form e ∪ f ∈ Ext1(B#C,A) andf ∪ e ∈ Ext1(A#C,B). Then the corresponding extensions are isomorphic:

A#(B#C) ∼= B#(A#C).

Similarly we have

A#(B#C) ∼= (A#B)#C,

if the classes in Ext1(B#C,A), Ext1(C,A#B) defining them map to the classes inExt1(B,A), Ext1(C,B) defining A#B, B#C respectively.

Proof. The first assertion follows from the diagram of triangles

B

B

A // A#(B#C) //

B#C

A // A#C // C .

12 R. P. THOMAS

Here we start with the bottom two rows and left and right hand columns, and thisdefines the arrow A#(B#C) → A#C by taking cones. The octahedral Lemma thengives the top row, so the central column now shows that A#(B#C) ∼= B#(A#C).

The second statement follows from similar yoga around the diagram

A // A#B //

B

A // E //

B#C

C C

In our use of this Lemma below, # will be the unique nontrivial extensionbetween the objects concerned. We use the notation ∼= for quasi-isomorphism andC [−n] for a copy of C shifted into degree n, so that Hom(E,F ) ∼= C [−n] is equivalentto Hom∗(E,F ) = C for ∗ = n and zero otherwise. The following is best interpretedin terms of pictures such as Figure 3 and the discussion of graded connect sums andrelative connect sums in [Th], [TY].

Proposition 5.2. Define Pii := Pi, then inductively (on j ≥ i) one can definePi,j+1 := Pij#Pj+1 by Hom1(Pj+1, Pij) = C.

Moreover we then have Hom(Pk, Pij) ∼=

C [1−N ] k = i− 1C k = iC [−N ] k = jC [−1] k = j + 1

and zero otherwise.

Proof. The result is true for Pii, i.e. j = i, by the fact that the Pi form an ANk -chain

of spherical objects (3.1). Inductively then, assume it is true for j, so that we havedefined Pi,j+1 = Pij#Pj+1.

Then the trianglePij → Pi,j+1 → Pj+1 → Pij [1]

gives

(5.3) Hom(Pk, Pij) → Hom(Pk, Pi,j+1) → Hom(Pk, Pj+1) → Hom(Pk, Pij)[1].

• For k < i− 1, k > j+2, or i < k < j, the second complex is (quasi-isomorphic to)zero since all the others are by the induction assumption.• For k = i− 1 (respectively k = i) (5.3) becomes

C [−d] → Hom(Pk, Pi,j+1) → 0 → C [1− d],

where d = N − 1 (respectively d = 0), so that Hom(Pk, Pi,j+1) ∼= C [−d] as required.• When k = j (5.3) becomes

C [−N ] → Hom(Pj , Pi,j+1) → C [1−N ] → C [1−N ].

We claim this last map HomN−1(Pj , Pj+1) → HomN (Pj , Pij) is an isomorphism, sothat Hom(Pj , Pi,j+1) ∼= 0 as required. This follows from the fact that the composition

to HomN (Pj , Pj) = C, is, by construction, the Yoneda product

HomN−1(Pj , Pj+1)⊗Hom1(Pj+1, Pj) → HomN (Pj , Pj),


which is an isomorphism by the duality (3.3).• For k = j + 1 we get

C [−1] → Hom(Pj+1, Pi,j+1) → C⊕ C [−N ] → C,

with the Hom0(Pj+1, Pj+1) → Hom1(Pj+1, Pij) component of the last map takes theidentity to the extension class defining Pi,j+1 = Pij#Pj+1. Since this was chosen tobe nontrivial, this map is an isomorphism, so Hom(Pj+1, Pi,j+1) ∼= C [−N ].• Finally, when k = j + 2, we have by (5.3)

0 → Hom(Pj+2, Pi,j+1) → C [−1] → 0,

as required. In particular, this now allows us to define Pi,j+2 and continue the in-duction.

Thus Hom1(Pij , Pi−1) = C and we can form Pi−1#Pij , which by inductive useof Lemma 5.1 is Pi−1,j . (Alternatively, Hom0(Pi−1, Pi−1,j) = C with cone Pij givesthe same result, as the extension cannot be trivial by the simplicity of Pi−1,j demon-strated in Proposition 5.7 below.) More generally, there is a unique extensionPij#Pjk

∼= Pik (again see Proposition 5.7, for instance). That is, we may writePij = Pi#Pi+1# . . .#Pj without confusion.

P3P4

P1#P2

P1#P2#P3

P1#P2#P3#P4

P3#P4

P1

P2

Figure 3. Our collection of stable objects

To define a stability condition on our category DNk , we need to define Z(Pi) for

all i, and the set of (semi)stable objects. We assume that N ≥ 2, so that there areno nontrivial extensions between any Pi and itself. Fix any sequence of positive realnumbers mi, and a sequence of real numbers

(5.4) φ1 < φ2 < . . . < φk < φ1 + 1.

Given any two integers i ≤ j, we define Pij := Pi#Pi+1# . . .#Pj as in theProposition above, and let φij be the unique real number in the interval [φi, φj ] ⊆

[φ1, φ1 + 1) such that mieiπφi + . . . + mje

iπφj = mijeiπφij for some positive real

numbers mij . Note this gives us inequalities

(5.5) φij < φkl if i < k and j < l, and φij ∈ [φ1, φ1 + 1) ∀ij.

14 R. P. THOMAS

Definition 5.6. Fix N ≥ 2 and mij, φij as above. Define

Z(Pi) := mieiπφi ,

and extend Z to be defined on all of K(Dk) = ⊕i ZPiby linearity, so that Z(Pij) =

Z(Pi) + . . .+ Z(Pj) = mijeiπφij .

Then define the Pij [m]s to be the stable objects of phase φij + m in DNk , and

define the SS(φ)s to consist of all direct sums of the stable objects of phase φ.

To show this is a stability condition we need to understand the Homs betweenthese Pijs, which we do now using Proposition 5.3.

Proposition 5.7. For i < k < j + 1 < l + 1 we have

(5.8) Hom(Pkl, Pij) ∼= C [−1]⊕ C [−N ],

while if one of the inequalities becomes an equality we get only one of the two sum-mands: for i = k < j +1 < l+1 and i < k < j +1 = l+1 we get C [−N ], while fori < k = j + 1 < l + 1 we get C [−1].

The duality Hom(Pkl, Pij) ∼= Hom(Pij , Pkl)∨[−N ] (3.3) applied to (5.8) deter-

mines more Homs. All others are zero apart from Hom(Pij , Pij) ∼= C⊕C [−N ]: thePij are spherical.

Proof. Building up Pij = Pi,j−1#Pj = Pi#Pi+1,j inductively and using Proposition5.3 gives this result very easily. We give the example of most interest to us: i < k =j + 1 < l + 1.

Using Hom(Pr, Pij) ∼= 0 for r > j + 1 (5.3), it is easy to show inductively thatHom(Prl, Pij) = 0 for r > j + 1 (where if l < r we define Prl := 0). Then applyingHom( · , Pij) to

Pj+1 → Pj+1,l → Pj+2,l

gives (5.3)C [−1] → Hom(Pj+1,l, Pij) → 0 → C.

Thus Hom(Pj+1,l, Pij) = C [−1] as required.The fact that the Pij are spherical is also proved inductively from the observation

that if A and B are spherical with Hom(B,A) = C [−1], then the correspondingextension A#B is also spherical (the connect sum of two spheres is a sphere!): fromthe triangle A→ A#B → B we get

Hom(B,A) //

Hom(B,A#B) //

Hom(B,B)

Hom(A#B,A) //

Hom(A#B,A#B) //

Hom(A#B,B)

Hom(A,A) // Hom(A,A#B) // Hom(A,B).

In the first column the connecting map Hom0(A,A) → Hom1(B,A) takes the iden-tity to the generator, by definition of the nontrivial extension. Since A is spherical,this makes Hom(A#B,A) ∼= C [−N ]. Similarly with the last column, using thefunctorial duality Hom(E,F )∨ ∼= Hom(F,E)[N ]. So the central row becomes

C [−N ] → Hom(A#B,A#B) → C,


from which it follows that A#B is spherical, since N ≥ 2.

Finally we can prove that we are in the situation of Theorem 4.3.

Theorem 5.9. Definition 5.6 defines a stability condition (4.1) on DNk .

Proof. Axioms (a), (b) and (e) of (4.1) are immediate from Definition 5.6, while (d)follows from Proposition 5.7 and the inequalities (5.5). Axiom (c) will follow fromTheorem 4.3 if we can show that any nontrivial extension E → C → F , with E, Fstable of phases φ < ψ, C is either stable (of phase θ ∈ (φ,ψ)), or a sum of stablesC = A⊕B, with A ∈ S(α), B ∈ S(β), φ < α ≤ β < ψ.

Such extensions are given by the shifts of the Homs computed in Proposition5.7. Only the Homs listed there in degrees 0 and 1 interest us: if there is a nonzeroHomn(F,E) with n ≥ 2 and the phases of E and F less than 1 apart (5.5) then thisgives rise to an extension in Hom1(F,E[n − 1]) with the phase of E[n − 1] greaterthan the phase of F , which therefore does not concern us. Note, however, that byduality, Homs of degree N, N − 1 give rise to Homs of degree 0, 1, in the oppositedirection, that we do need to consider. This gives us 5 cases to check (there are alsoselfHoms to consider, but these just give P#P [1] ∼= 0).• Pij#Pj+1,l, and their shifts. We have already observed that this gives the singlestable object Pil with phase φil ∈ (φij , φj+1,l).• Pkl#(Pkj[1]), j < l, and their shifts. This extension comes from the element of

Hom0(Pkj , Pkl) in the triangle Pkj → Pkl → Pj+1,l, and so is isomorphic to Pj+1,l,with phase φj+1,l ∈ (φkl, φkj + 1).• Pkl#(Pil[1]), i < k, and their shifts. This extension comes from the element ofHom0(Pil, Pkl) in the triangle Pi,k−1 → Pil → Pkl, and so is isomorphic to Pi,k−1[1],with phase φi,k−1 + 1 ∈ (φkl, φil + 1).

• Pij#Pkl, i < k < j + 1 < l + 1, and their shifts. This element of Hom1 is in theimage of Hom1(Pkl, Pi,k−1) (that defines Pi,k−1#Pkl = Pil) via the map Pi,k−1 → Pij

(with cone Pkj). So we induce a diagram

Pi,k−1 //

Pil//

Pkl

Pij //

Pij#Pkl //

Pkl

Pkj Pkj

where the bottom row is induced from the octahedral Lemma. So the central columnshows that Pij#Pkl

∼= Pkj ⊕Pil, as there are no nontrivial extensions between thesetwo objects, by Proposition (5.7). (See also Figure 4.) The inequalities (5.5) givethe required φkj , φil ∈ (φij , φkl).

16 R. P. THOMAS

• Pkl#(Pij [1]), i < k < j + 1 < l + 1, and their shifts. Similarly the diagram

Pij

Pij

Pkj //

Pkl//

Pj+1,l

Pi,k−1[1] // Pkl#(Pij [1]) // Pj+1,l

of the standard morphisms we have already seen shows that Pkl#Pij[1] ∼= Pj+1,l ⊕Pi,k−1[1] since there are no extensions between these two objects. Again (5.5) showsthat φj+1,l, φi,k−1 + 1 ∈ (φkl, φij + 1).

Pij#PklPij Pkl

Pij

Pj+1,l

Pkl Pij#Pkl∼= Pil ⊕ Pkj

Pkj

Pij

Pil

Pkl

Pi,k−1[1]

Pkl#Pij [1] ∼= Pi,k−1[1]⊕ Pj+1,l

Figure 4. Connect sums of stable objects Pij and Pkl of increasingphase are (direct sums of) stable objects: they (the curved lines) canbe deformed to the dashed straight lines.

6. Deforming the stability condition

In this section we determine an entire connected component of the space ofstability conditions in dimension N = 2 and connect it with braid groups of autoe-quivalences [RZ], [KS], [ST]. The proofs are a little brief; in particular they makeuse of the description of stability conditions mentioned in (4.7) and proved in [Br1].

In [Br1] the space of stability conditions is shown to be a metric space in anatural way, such that the mass function (4.2) is continuous. Using the descriptionof stability described in the paragraph (4.7), the space is shown locally, about anygiven stability condition, to be isomorphic to the space Hom(K(T ),C) of Zs. Infact it is shown to be a cover of the space of Zs minus those where a mass (4.2)goes to zero. That is the stability condition deforms even through walls of Zs wheresome objects become unstable and others stable, so long as Z of a stable object doesnot become zero. We will show that in our case this means that the points 0 and

Z(P1j) =∑j

i=1 Z(Pi) are distinct, i.e., geometrically, the “endpoints” of our specialLagrangians are distinct – no stable object’s mass has gone to zero. On loopinground such a zero (a generator of the braid group) back to the same Z, we willfind that the stability condition has changed, as the set of stable objects undergoesa “Dehn twist” [ST], at least in the easiest case to analyse, dimension N = 2, towhich we will restrict from now on.


To do this we will need show that under deformations of Z, our set of stableobjects keep the following properties of our initial set of stable objects (5.6).

Definition 6.1. The set of stable objects of a stability condition on D2k is called

simple if it is the set of shifts of k(k + 1)/2 distinct spherical objects Qij , 1 ≤ i ≤j ≤ k, satisfying the following conditions. [Qij ] = [Pij ] in K-theory, there is a singleHom in some degree between Qab and Qb+1,c, a single Hom in some degree betweenQab and Qac (c 6= a), and the Euler characteristic

(6.2) χ(Qab, Qcd) :=∑

i

(−1)i dimHomi(Qab, Qcd) = 0 for a < c 6= b+ 1.

We call the stability condition simple if its set of stable objects is simple and itssemistable objects are direct sums of stable objects of the same phase.

We want to show next that the stability condition remains simple on passingthrough walls of semistability.

We say that Z ∈ Hom(K(Dk),C) lies on a codimension one wall if Z([Pij ]) 6=0 ∀i, j, at least 2 of the classes [Pij ] have the same phase (mod 1), and at most 3, inwhich case the three must be linearly dependent (and so of the form [Pab], [Pb+1 c], [Pac]– notice that for the sum of two [Pij ] classes to equal the class of another, the twoclasses must be of the form [Pab], [Pb+1 c].)

We can talk locally about “sides” of such a wall depending on the sign of thedifference in sign of these two phases.

Proposition 6.3. If P, Q are distinct stable objects in a simple stability condi-tion, close to, and on one side of, a codimension one wall, and whose phases coincideon the wall, then the total dimension of Hom∗(P,Q) is at most 1.

If the total dimension of Hom∗(P,Q) is exactly 1, then there is a further stableobject R of the same phase on the wall, and, on reordering P,Q,R if necessary, theHom is of degree 1 and R ∼= P#Q.

Proof. We need to rule out there being two or more Homs between P and Q. In thiscase, by the definition of codimension one wall and simple (6.1), there are no otherK-theory classes of stable objects whose phases tend to those of [P ], [Q] on the wall.

Shifting Q and swapping P, Q if necessary, and moving closer to the wall, wecan assume that the phase of Q is more than that of P , and that there are no stableobjects of phase in between. Thus, by stability and Serre duality, Homi(Q,P ) = 0for i ≤ 0 and i ≥ 3. By the vanishing of the Euler characteristic (6.2), then, theremust be equal numbers of Homs in degrees 1 and 2; pick a nonzero element ofHom1(Q,P ) and form the extension P#Q. Using the description of stability givenin the paragraph (4.7), if P#Q were unstable there would be a stable object, with anonzero Hom to P#Q, of phase between those of P, Q. But this is a contradiction,so that P#Q is stable – another contradiction, since its K-theory class is [P ] + [Q],which does not contain a stable object.

So we need only consider the case where there is a single Hom from Q to P ,which by axiom (d) and Serre duality must be in degree 1 or 2 (for the phase ofQ greater than that of P ). There are now 3 stable objects P,Q,R whose phase is

18 R. P. THOMAS

tending to the same value on the wall. We work with the semistability criterionin the abelian category (4.7) (for a suitable value of t so that it contains P,Q,R).Without loss of generality we will assume that the masses (4.2) of P,Q are less thanor equal to that of R on the wall.

There can be no element of Hom2(Q,P ) ∼= Hom0(P,Q)∗, as the image in Q ofany Hom from P (i.e. the cokernel of its kernel in the abelian heart) would havea filtration by stable objects of phase between those of P and Q, and of strictlysmaller mass.

So Hom1(Q,P ) = C, and we can form P#Q. Its Harder-Narasimhan filtrationin the abelian category (4.7) is of semistable objects of phase between those of Pand Q; i.e. of direct sums of shifts of P,Q,R. Since the extension is non-trivial andthe masses of P,Q are less than or equal to that of R, it follows that the filtrationis just R, that is P#Q = R. We claim that there is a single Hom between R andeither of P and Q; for instance Hom(R,P ) fits into long exact sequence

Homi(Q,P ) → Homi(P#Q,P ) → Homi(P,P ) → Homi(Q,P ) → . . . ,

in which the identity in Hom(P,P ) maps to the generator of Hom1(Q,P ). ThusHom(P#Q,P ) ∼= C [2]. Similarly Hom(Q,P#Q) ∼= C [2].

Proposition 6.4. On crossing a codimension one wall a simple stability condi-tion remains simple.

Proof. Suppose the phases of two stable objects P and Q (on one side of the wall,where the phase of P is less than that of Q, without loss of generality) coincide onthe wall. If there are no Homs between them it is easy to see that there are no othersuch stable objects, and the set of (semi)stable objects does not change across thewall; in particular P and Q remain stable (see for example [Br1] for more details,or use the equivalent definition of stability in the paragraph (4.7)).

So by Proposition 6.3 we need only consider the case where there is a single Homfrom Q to P , of degree one, and R = P#Q is also stable.

Then for P,Q to become unstable as we cross the wall, they must have a filtrationby stable objects of strictly smaller mass and the same phase on the wall. Butbecause this is a codimension one wall and R has greater mass, no such objectsexist.

R becomes unstable on the other side of the wall. But there we can use Serreduality to form a unique nontrivial extension Q#P , which is similarly stable on thatside of the wall as in the proof of Proposition 6.3. Q#P is also spherical as in theproof of Proposition 5.7. Since no other stable objects are affected we claim thatthis new set of stable objects (with each shift P#Q[r] replaced by Q#P [r]) is alsosimple. The K-theory class of Q#P is the same as that of P#Q, and its Homs to Pand Q are one dimensional as in the last paragraph of the proof of Proposition 6.3.To satisfy Definition 6.1, then, we must finally check that for any stable object Ewith χ(E,P#Q) = 0 (6.2), we also have χ(E,Q#P ) = 0. But by the exact sequenceHomi(E,Q) → Homi(E,Q#P ) → Homi(E,P ) → Homi+1(E,Q) → . . . we see that

χ(E,Q#P ) = χ(E,Q) + χ(E,P ) = χ(E,P#Q) = 0,


as required.

While we have analysed crossing only codimension one walls, the fact from [Br1],mentioned above, that locally the space of stability conditions is isomorphic to thespace of Zs means that there is no monodromy around codimension 2 walls, andwhenever Z does not lie on a wall, the stability condition is simple. In particular,then, there is always a stable object in the K-theory class [Pij ] away from thefinite number of walls. Combined with the results on deforming stability conditions[Br1], which can always be done until a stable object’s mass goes to zero, we findthe connected component of our stability conditions (5.6) is a cover of the space ofZs such that Z([Pij ]) 6= 0 for all i ≤ j. Plotting the points 0 and Z([P1i]) for all i,and translating them to have mean zero, this space in turn covers the configurationspace C0

k+1 of (k + 1) distinct points in C with centre of mass the origin. We nowprove slightly more.

Theorem 6.5. Via the above map, the connected component of the stabilityconditions (5.6) is the universal cover of the configuration space C0

k+1. The decktransformations are given by the Bk action of [KS], [ST].

Proof. We want to check that the result of going round loops in configuration spaceis the braid group action of [KS], [ST]. It is sufficient to check this for a suitablechoice of generators; namely we pick a stability condition as in (5.6) in which themass |Z(Pii)| of the stable object Pi is strictly smaller than that of all other stableobjects, and we move Z by rotating Z(Pi) anticlockwise through π radians whilefixing Z of stable objects of different endpoints. As we rotate Z(Pi), the phases ofthe stable objects P with an endpoint (i− 1) or i will also change (by no more thanǫ = sin−1(|Z(Pi)|/|Z(P )|), by the triangle inequality); we also assume that |Z(Pii)|is so small that no two phases of classes [Pkl] 6= [Pi] coincide under this rotation; i.e.we take all the phases of the classes [Pkl] to be distinct, and ǫ to be smaller thanthe smallest difference in such phases.

Note that for this choice of stability condition and stable object Pi, there areno other stable objects with 2 Homs to Pi (5.3); they either have no Homs or one.As we rotate Z(Pi) and follow the stability condition below, we can see that thisproperty is preserved, as by design no stable objects’ phases become equal, exceptto the phase of Pi, so no stable objects change except via their interaction with Pi

described now.As we rotate Z(Pi) and cross walls (of codimension one only, without loss of

generality, by perturbing the loop if necessary), stable objects with no Homs areunaffected, while those with one Hom are altered as in the proof of the last Propo-sition.

That is, as the phase of Z(Pi) reaches that of some other pair of stable objectsE,F (F of smaller mass) with Hom(Pi, E) ∼= C [0], Hom(Pi, F ) ∼= C [−1] and E ∼=Pi#F . On passing through the wall the stable object E is replaced by F#Pi, andall other stable objects are left unchanged.

But we claim that

TPi(Pi#F ) ∼= F and TPi

F ∼= F#Pi,

20 R. P. THOMAS

where TPiis the Dehn twist about the spherical object Pi; an equivalence of trian-

gulated categories [ST]. TPi(E) sits in the triangle

Pi ⊗Hom(Pi, E) → E → TPi(E),

where the first map is evaluation. Since Hom(Pi, Pi#F ) ∼= C [0] and Hom(Pi, F ) ∼=C [−1], applying this to E = Pi#F and E = F gives the standard triangles

Pi → Pi#F → F and Pi[−1] → F → F#Pi

respectively, proving the claim.So on listing the stable objects in ascending phase, we find that the subsequence

Pi, E, F is replaced by F,F#Pi, Pi, i.e. by TPiE,TPi

F,Pi.The same is true when the phase of Pi passes through that of the other stable

objects P with no Homs to Pi: they are left unaffected, just as under TPi: TPi

P ∼= P .As we rotate Z(Pi) through π, its phase crosses the phase of all stable objects

(after a suitable shift), and so we end up with the list of stable objects of ascendingphase Pi, A,B, . . . being replaced by TPi

A,TPiB, . . . , Pi. Equivalently, as Pis phase

has increased by 1, we have Pi[−1], TPiA,TPi

B, . . .. But TPiPi

∼= Pi[−1], so we havealtered the set of stable objects (and Z) by the action of TPi

, as claimed.Finally, since the Bk action of ([KS], [ST]) is faithful not just on the triangu-

lated categories, but also on Ak-chains such as the stable objects Pi ([ST] Theorem4.13), we see that the cover of Ck+1 that we get is the universal cover.

We end by noting that the if we take an element of the braid group whichacts trivially on K-theory (and so on Z), then the two stability conditions differonly by their set of stable objects. In our case, these stable objects differed byan autoequivalence of the triangulated category. More generally, if the axiomaticnotion of stability of [Br1] is to agree with the physical notion of stability (andthe geometric conjectures in [Th]) this would have to hold more generally; that isone might conjecture that two stability conditions with the same central charge ona “Calabi-Yau category” (one with a “trace map” HomN (E,E) → C inducing afunctorial duality Hom(A,B)∗ ∼= Hom(B,A)[N ] for all A,B) should differ by anautoequivalence of the category that is the identity on its numerical K-theory.

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Department of Mathematics, Imperial College, 180 Queen’s Gate, London SW72BZ. UK.Email: [email protected]

arXiv:math/0212214v5 [math.AG] 26 Apr 2006

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